It is also between 10 and 127, there is an unstated assumption that the question means "consecutive integers". Of course the question may have said exactly that, but you did not think it important enough to mention?

I also know that and that . Since 150 lies between 144 and 169, its square root lies between 12 and 13.

As Captain Black pointed out, You can pick any integer whose square is less than 150 and any (positive) integer whose square is larger than 150 and say that the square root of 150 lies between them. But because you specifically asked "what two integers", it was clear that you wanted a single answer and the most reasonable interpretation of that is the two integers closest to that square root.